3 Answers
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A chord does not have to be made up of thirds. A chord is by definition two or more notes heard as if sounded simultaneously. Not all chords have three notes either. There are dyads (two notes), triads (three), tetrachords (four), pentachords (five), and hexachords (six). There's no limit on the number of notes, and also, by definition, there's no limits on which notes. C - E - G is a chord. D - E - F - C is a chord. However, the most common triads are the major, minor, augmented, and diminished (there is also the suspended). All of these are composed of a root, a third, and a fifth (except the suspended, which uses the root, perfect fourth and perfect fifth).

So, now to your question, why thirds? First, realize there are two types of thirds: the major and the minor. The major consists of four semitones and the minor three semitones. Quoting Wikipedia:

The major third is classed as an imperfect consonance and is considered one of the most consonant intervals after the unison, octave, perfect fifth, and perfect fourth. In the common practice period, thirds were considered interesting and dynamic consonances along with their inverses the sixths.

After the major third became established as such, it become pretty standard. Every classical piece makes use of it in some way. The other reason the major third is so widely used is that it is found in the harmonic series (between the fourth and fifth). Early brass (e.g., posthorn, natural trumpet) had no valves or slides and were limited to the harmonic series. This encouraged use of and familiarity with the major third. However, I'd say the most important of all these reasons is the first. It is highly consonant.

The minor third has the same level of consonance as the major third, but is found higher up in the harmonic series (between fifth and sixth). Also, there are many common transposing instruments which sound a minor third higher or lower form where they are written. For example, the Eb clarinet and the Eb trumpet both sound a minor third higher than written. The oboe d'amore, popular in the eighteenth and twentieth centuries, and the soprano clarinet in A sound a minor third higher than written. Of these reasons, I'd say the first (again) is the most important.

As for other intervals, any interval can be used, but some are more common than others. The perfect fifth, octave, unison, and seventh (in no particular order) are very common. All major, minor, and suspended chords have a perfect fifth. Also important are the perfect second, perfect fourth, and major sixth. To learn more about different chords and the intervals that make them up, read this article on intervals and this one on chords. They are both very informative.

TL;DR

Why thirds?

Thirds are the most consonant intervals (after the unison, octave, perfect fifth, and perfect fourth).

Are other intervals sometimes used?

Many other intervals are used. See here for a list of the main ones. They include the perfect fifth, the perfect seventh, the octave, the major sixth and the perfect fourth.

It might be useful to inject a note about the language we use when talking about music, and specifically music theory. It sounds like you are asking about why music theory would call one thing a chord and not some other thing.

"Building" and "constructing" have no precise meaning when used in music. Are you talking about "building" an actual chord that gets played in a piece? Or are are you thinking more of a "blueprint" model; as in, what are the possible chords I could build, whether I eventually play one or any of them?

To help explain the difference, think of a "vertical slice" of a piece of music. As in, what happens in the half-second interval beginning at minute 1:31 of whatever song you are currently listening to. No matter what the music is, you are hearing what sometimes get called a "simultaneity." It's just the combination of "all" the sounds that are happening simultaneously.

If you're thinking of a solo piano piece, it might be that the piano has just struck three notes, and those are primarily the ones ringing out in this half-second sample you're imagining. But if you are thinking of an orchestral piece, it could be a hundred notes all being played simultaneously.

In both cases, it's not wrong to think of what you are hearing as a chord, but most people would tend to think only of the first case as the response to what you are asking. Though it might help you get your bearings to learn only chords "built" of thirds, you'll may eventually realize that any combination of notes may actually sound good in one moment in a piece.

The theory you are learning is describing what we would call "tertian harmony." In this case, you are "stacking" thirds. But what happens when you strike the open strings of a guitar in standard tuning? Why wouldn't we call that a chord, too? And this is where the semantic problems rear their ugly head. There's no reason not to call what you've just played a chord, but using the language of "tertian harmony" makes it a little complicated. Do you like the name "E minor 7th sus 4?" I sure don't, but that would be an accurate name (another, less precise but no less correct way to think of the open-strings chord is that it has a "quartal" sound- this means made from primarily stacking fourths).

Learning the taxonomy for naming vertical slices of notes is not as easy as saying C-E-G is a triad, but at least you're not stuck thinking you are limited in your choices.

Here's two famous examples re: "chord constructing"

Think about the amount of time spent devoted to thinking about something that takes about a send to hear:

This is the ending of "A Day in the Life", which is mentioned in the above article. Think about how you would describe the orchestral sounds that precede the big, final concluding E? Hard to describe, but no less powerful.

OK, here's the real answer, which hardly any theory texts explain properly and I only learnt when doing special studies in microtonality at music school.

Western harmony is derived from what Harry Partch called the five limit. You get your triadic chords by making intervals with fractions that have a denominator of less than five. Just tuned intervals below the five limit are (in order of consonance to dissonance): 2/1 octave, 3/2 - p5th, 4/3 - p4th, 5/4 - M3rd, 6/5 - min 3rd. To derive the notes in major harmony, you build a triad using 3/2 and 5/4 on the 1st, 4th and 5th. In other words, you stack 3/2 and 5/4 over 3/2 and 4/3. This gives you all the notes in the major scale. To do the same for minor harmony, you stack 3/2 and 6/5 over 3/2 and 4/3, giving you 1,2,b3,4,5,b6,b7. If you mix both, you have all the notes in western harmony less the tritone. So triadic harmony is actually far from arbitrary in it's pure form.

Partch's harmonic system was based on extending this by experimenting with higher limits and is one of the more interesting 20th century directions in harmony.